Strong approximation theorems for density dependent Markov chains

This review article is concerned to provide a global context to several works on the fitting of continuous time nonhomogeneous Markov chains with finite state space and also to point out some selected aspects of two techniques previously introduced—estimation and calibration—relevant for applications and used to fit a continuous time Markov chain model to data by the adequate selection of parameters. The denomination estimation suits the procedure better when statistical techniques—e.g., maximum likelihood estimators—are employed, while calibration covers the case where, for instance, some optimisation technique finds a best approximation parameter to ensure good model fitting. For completeness, we provide a short summary of well-known important notions and results formulated for nonhomogeneous Markov chains that, in general, can be transferred to the homogeneous case. Then, as an illustration for the homogeneous case, we present a selected Billingsley’s result on parameter estimation for irreducible chains with finite state space. In the nonhomogeneous case, we quote two recent results, one of the calibration type and the other with more of a statistical flavour. We provide an ample set of bibliographic references so that the reader wanting to pursue her/his studies will be able to do so more easily and productively.

Continuous time Markov chain based website navigability measure

The effects of a periodic contact rate and of carriers are considered for a generalization of Bailey's simple epidemic model. In this model it is assumed that individuals become susceptible again as soon as they recover from the infection so that a fixed population can be divided into a class of infectives and a class of susceptibles which vary with time. If the contact rate is periodic, then the number of infectives as time approaches infinity either tends to zero or is asymptotically periodic depending on whether the total population size is less than or greater than a threshold value. The behavior for large time of the number of infectives is determined for three modifications of the model which involve carriers.

Comparison of deterministic and stochastic SIS and SIR models in discrete time

The origins of modern spatial analysis lie in the development of quantitative geography and regional science in the late 1950s. The use of quantitative procedures and techniques to analyse patterns of points, lines, areas and surfaces depicted on analogue maps or defined by co-ordinates in two-or three-dimensional space characterise the initial stage. Later on, more emphasis was placed on the indigenous features of geographical space, on spatial choices and processes, and their implications for the spatio-temporal evolution of complex spatial systems. Spatial analysis, as it has become over the past four decades, is more than spatial statistics and data analysis, and goes far beyond data sampling, data manipulation, exploratory and confirmatory spatial data analysis, into areas of spatial modelling encompassing a large and diverse set of models in both the environmental and the social sciences. In the environmental sciences, they range from physical dispersion models (e.g. for suspended particulates), chemical reaction models (e.g. photochemical smog), and biological systems models (e.g. for ecosystems in water), through deterministic process and stochastic process models of regional environmental quality management, and integrated models for environment-energy-economic assessment. For many models, especially the physical dispersion models, chemical reaction models, and biological systems models, the geographical location of the site is not considered to be of overriding importance. The variation in model results over space can be achieved by obtaining different inputs to the area in question. Process models include deterministic versions which attempt to describe a particular sequence in terms of known physical laws, and stochastic model versions which aim to describe a particular process such as erosion, groundwater movement and absorption of pollutants in terms of probability theory. For such processes, there is an interaction between the spatial process and a substrate. This substrate provides a one-, two-or three-dimensional framework within which the process model can operate. The space in which the process is modelled is disaggregated into a set of finite elements which are usually assumed to be internally homogeneous. Two-dimensional models generally use as their data small pixels (see Fischer et al., 1996).

An Introduction to Markov Chain Monte Carlo

Hypersonic flow over Stardust Re-entry Capsule using ab-initio based chemical reaction model

We consider the susceptible-infective (SI) epidemiological model, a variant of the Kermack-McKendrick models, and let the contact rate be a function of the number of infectives, an indicator of disease spread during the course of the epidemic. We represent the resultant model as a continuous-time Markov chain. The result is a pure death (or birth) process with state-dependent rates, for which we find the probability distribution of the associated Markov chain by solving the Kolmogorov forward equations. This model is used to find the analytic solution of the SI model as well as the distribution of the epidemic duration. We use the maximum likelihood method to estimate contact rates based on observations of inter-infection time intervals. We compare the stochastic model to the corresponding deterministic models through a numerical experiment within a typical household. We also incorporate different intervention policies for vaccination, antiviral prophylaxis, isolation, and treatment considering both full and partial adherence to interventions among individuals.

A stochastic model of tumor growth incorporating several key elements of the growth processes is presented. Generalizing a previous work by the authors, two one-dimensional diffusion processes representing populations of proliferating and quiescent cells are obtained. Their forms turn out by their relation with total tumor population analysed in (1). The proposed model is able to incorporate the effects of the mutual interactions between the two subpopulations. It is also used to simulate the effects of two kinds of time-dependent therapies: non-specific cycle and specific cycle drugs. Moreover, the first-exit-time problem is analyzed to study cancer evolution in the presence of a time-dependent therapy. 1 Introduction and background Tumor is one of the main causes of death in our so- ciety so, in the last twenty years a lot of attempts have been made to describe the tumor kinetic. Cancer cells population consists of a combination of proliferating, quiescent and dead cells that determine tumor growth based on surrounding environmental conditions (cf. (5)). Furthermore, since experimental data show the existence of more or less intense random fluctuations in tumor growth, in a previous work (see (1)) the authors provided a stochastic generalization of the Gompertz law in order to model monoclonal tumor growth. So tumor size is described by means of a one-dimensional diffusion process X(t) and the first exit time problem (FET) for X(t) from a region D has been analysed. In particular, D is restricted by two absorbing boundaries representing healing threshold and the carrying capacity. A first natural generalization consists of including all the essential biological phenomena of a cellular population. To this aim, following Kozusko and Bajzer (cf. (7)), we split tumor population in two subpopulations: proliferating and quiescent cells. In this direction, in (7), the authors proposed a deterministic model to describe tumor dynamics, assuming that the transition rates between proliferating and quiescent populations depend on the total population size. By imposing that the total population is governed by the Gompertz law, Kozusko and Bajzer resolved analytically the model and obtained the dynamics of prolif- erating and quiescent populations as functions of the whole population size. Following this approach, we describe proliferating and quiescent populations via two new diffusion pro- cesses, generally time-non-homogeneous, connected to the process X(t). The FET problem for such processes through suitable boundaries is considered. The followed approach per- mits to analyse the effect of different therapies on tumor cells. In Section 2 we will review the main characteristics of the model proposed in (1) in which the effect of a therapy is seen as a moderation term of the growth rate of the tumor cells. In Section 3 we generalize the deterministic model by Kozusko and Bajzer by obtaining two diffusion processes P(t) and Q(t) representing the populations of proliferating and quies- cent cells, respectively. Finally, in Section 4 the effect of a therapy on X(t) is investigated

We propose a stochastic logistic model with mate limitation and stochastic immigration. Incorporating stochastic immigration into a continuous time Markov chain model, we derive and analyze the associated master equation. By a standard result, there exists a unique globally stable positive stationary distribution. We show that such stationary distribution admits a bimodal profile which implies that a strong Allee effect exists in the stochastic model. Such strong Allee effect disappears and threshold phenomenon emerges as the total population size goes to infinity. Stochasticity vanishes and the model becomes deterministic as the total population size goes to infinity. This implies that there is only one possible fate (either to die out or survive) for a species constrained to a specific community and whether population eventually goes extinct or persists does not depend on initial population density but on a critical inherent constant determined by birth, death and mate limitation. Such a conclusion interprets differently from the classical ordinary differential equation model and thus a paradox on strong Allee effect occurs. Such paradox illustrates the diffusion theory's dilemma.

We treat a model describing the continued formation and growth of mutant biological populations. At each transition time of a Poisson process a new mutant population begins its evolution with a fixed number of elements and evolves according to the laws of a continuous time positive recurrent Markov Chain $Y(t)$ with stationary transition probabilities $P_{ik}(t), i, k = 0,1,2,\cdots, t \geqq 0$. Our principal concern is the asymptotic behavior of moments and of the distribution function of the functional $S(t) = \{$number of different sizes of mutant populations at time $t\}$. When the recurrence time distribution to any state of the Markov Chain $Y(t)$ has a finite second moment, the moments of $S(t)$ and limit behavior of its distribution function are controlled by the stationary measure associated with $Y(t)$. When the second moment of the recurrence time distribution is infinite, then a local limit theorem and speed of convergence estimate for $P_{ik}(t)$ with $k = k(t) \rightarrow \infty, t \rightarrow \infty$ are required to establish asymptotic formulas for moments of $S(t)$.

We study new formulae based on Lya- punov exponents for entropy, mutual information, and capacity of finite state discrete time Markov chan- nels. We also develop a method for directly com- puting mutual information and entropy using contin- uous state space Markov chains. We show that the entropy rate for a symbol sequence is equal to the primary Lyapunov exponent for a product of random matrices. We then develop a continuous state space Markov chain formulation that allows us to directly compute entropy rates as expectations with respect to the Markov chain's stationary distribution. We also show that the stationary distribution is a continuous function of the input symbol dynamics. This continu- ity allows the channel capacity to be written in terms of Lyapunov exponents.

Eight populations of Drosophila serrata were used in a factorial design of the type 2 x 2 x 2 to study the influence of some environmental and genetic factors on the productivity and total size of a population under conditions of severe competition. The variables studied were two temperatures (25 C versus 19 C), two feeding schedules (2 versus 3 food units per week), and two genetic compositions (single strain versus two strain hybrid population). The parameters measured were the number of flies, the biomass produced per food unit, and the total population size. The results show that the amount of food is the main factor controlling the number of flies and the biomass produced by a population. The higher temperature has a slight favorable effect on the number of flies, but not on the biomass produced; that is, at 25 C a few more flies are produced but they are lighter. When the flies oviposit in the bottle for a longer time, the number of flies and the biomass produced are increased. The main factor controlling the population size seems to be the living space. An increase of 50 % in the amount of food provided to the population only increases the population size slightly. At the lower temperature, there is an increase in the population size, possibly due to a decrease in the activity of the flies. The hybrid populations are considerably superior to the single strain populations, both in productivity and in total size, presumably owing to the greater genetic variability they offer to the action of natural selection

On the effects of migration and spatial heterogeneity on single and multiple species

Only in the last twenty years have ornithologists given serious attention to monitoring changes in bird populations. Seabirds, though sometimes presenting problems of access to their breeding sites, are unusual in generally offering opportunities to obtain estimates of their total population size. They usually display a clumped distribution often in sizeable colonies where there are stretches of suitable coastline or offshore islands, in contrast to most passerines that occur more or less evenly in a patchwork of suitable habitat. Despite these advantages there are very few seabird species (one example is the Northern Gannet Sula bassana) for which we can be confident of good estimates of their total population size. Instead we reply upon sampling populations over their natural range with the degree of effort varying greatly between regions (usually on a national basis) depending partly upon difficulties of access but mainly upon the resources available within that country. For most people, monitoring in this context means to follow changes in the numbers of birds or nests by frequent standardized counts. I should like to extend it to aim also for complete censuses at less frequent intervals (see below).